187 lines
5.8 KiB
TeX
187 lines
5.8 KiB
TeX
% TODO REMARK ABOUT ZORNS LEMMA (LECTURE 1)
|
|
|
|
% TODO REMARK ABOUT FIN PRESENTED MODULES (LECTURE 2)
|
|
|
|
|
|
% TODO: LECTURE 9 LEMMA
|
|
|
|
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% ÜBERSICHT %
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
% List of forms of HNS
|
|
|
|
|
|
\begin{itemize}
|
|
\item[HNS2 $\implies$ HNS1b]
|
|
Let $I \subseteq \mathfrak{l}[X_1,\ldots,X_n]$. $I \subseteq \mathfrak{m}$
|
|
maximal. $R / \mathfrak{m}$ is isomorphic to a field extension of
|
|
$\mathfrak{l}$.
|
|
Finite by HNS2.
|
|
\item[NNT $\implies$ HNS2]
|
|
Apply NNT to $L / K$ $\leadsto$ alg. independent $a_i$ such that $L$ is finite
|
|
over the image of $K[X_1,\ldots,X_n] \xrightarrow{\ev_a}
|
|
L$ ($\ev_a : P \to P(a_1,\ldots,a_n)$).
|
|
$\ev_a(K[X_1,\ldots,X_n])$ is a field by fact about integrality and fields
|
|
\ref{fintaf}.
|
|
Hence $n = 0$ and $L / K$ is finite.
|
|
\item[UNCHNS2]
|
|
$K$ uncountable, $L / K$ fin. type.
|
|
Then $\dim_K L$ is countable.
|
|
Suppose $l \in L$ is not integral.
|
|
Then $K(l) \cong K(T)$ and $\dim_K L \ge \dim_K K(T) \ge \aleph_1$.
|
|
Thus $L / K$ algebraic $\implies$ integral $\implies$ finite.
|
|
\item[HNS3]
|
|
($V(I) \subseteq V(f) \iff f \in \sqrt{I} $).
|
|
Suppose $V(I) \subseteq V(f)$.
|
|
$R' \coloneqq \mathfrak{k}[X_1,\ldots,X_n, T], J \subseteq R'$ the ideal generated by $I$
|
|
and $g(X_1,\ldots,X_n,T) \coloneqq 1 - Tf(X_1,\ldots,X_n)$.
|
|
\end{itemize}
|
|
|
|
|
|
% Proofs
|
|
Def of integrality (<=>)
|
|
|
|
|
|
Fact about integrality and field:
|
|
|
|
|
|
Technical lemma for Noether normalization: For $S \subseteq \N^n$ finite, there
|
|
exists $k \in \N^n$ such that $k_1 = 1$ and $s_1 \neq s_2 \in S \implies
|
|
\langle k, s_1 \rangle \neq \langle k, s_2 \rangle$: For $s_1 \neq s_2$,
|
|
|
|
Noether normalization: $a_i \in A$ minimal such that $A$ is integral over the
|
|
subalgebra genereted by the $a_i$.
|
|
% TODO% TODO
|
|
Suppose $\exists P \in K[X_1,\ldots,X_n] \setminus \{ 0\} ~
|
|
P(a_1,\ldots,a_n) =
|
|
0$.
|
|
$P = \sum_{\alpha \in \N^n} p_\alpha X^\alpha, S \coloneqq \{ \alpha \in \N^n
|
|
| p_\alpha \neq 0\}$.
|
|
Choose $k$ as in the lemma.
|
|
$b_i \coloneqq a_{i+1} -
|
|
a_1^{k_{i+1}}, 1 \le i <n$.
|
|
Claim: $A$ is integral ober subalgebra $B$ generated by the $b_i$ ($\lightning$
|
|
minimality) $Q(T) \coloneqq P(T, b_1 + T^{k_2}, \ldots,
|
|
b_{n-1} + T^{k_n})$.
|
|
$Q(a_1) = P(\vec a) = 0$.
|
|
For suitable $\beta_{\alpha, l} in B$:
|
|
\[
|
|
T^{\alpha_1} \prod_{i=1}^{n-1} (b_i +
|
|
T^{k_{i+1}})^{\alpha_{i+1}} =
|
|
T^{w_k(\alpha)} + \sum_{l=0}^{w_k(\alpha) - 1}
|
|
\beta_{\alpha,l} T^l
|
|
\]
|
|
Thus $Q(T) = p_\alpha T^{w_k(\alpha)} + \ldots$ where
|
|
$\alpha \in S$ such that $w_k(\alpha)$ is maximal.
|
|
Thus, $Q$ is normed.
|
|
|
|
|
|
% TODO Artin-Tate
|
|
|
|
|
|
|
|
%
|
|
A first result of dimension theory: $A \mathfrak{l}$-algebra of
|
|
finite type,
|
|
$\fp, \fq \in \Spec A, \fp \subsetneq \fq$.
|
|
Then $\trdeg(\mathfrak{k}(\fp) /\mathfrak{l}) > \trdeg(\mathfrak{k}(\fq) /
|
|
\mathfrak{l})$: Without loss of
|
|
generality loss of generality $\fp = \{0\}$
|
|
and $A$ a domain ($A' \coloneqq A / \fp$).
|
|
For $\fq \in \MaxSpec A$, $\mathfrak{k}(\fq) = A / \fq$ finite type, hence
|
|
finite (HNS) $\implies \trdeg(\mathfrak{k}(\fq) / \mathfrak{l}) = 0$.
|
|
$\trdeg(Q(A) / \mathfrak{l}) = 0 \implies A$ integral over
|
|
$\mathfrak{k}$ $\implies$ $A$ a field $\implies \fp = \fq
|
|
\lightning$.
|
|
|
|
If $\fq \not\in \MaxSpec A$, let $a_1,\ldots,a_n \in A$ alg.
|
|
independent such that the $\overline{a_i}$ are a transcendence base for
|
|
$\mathfrak{k}(\fq) / \mathfrak{k}$
|
|
Let $R$ be the ring generated by $\mathfrak{l}$ and the $a_i$.
|
|
Localize with respect to $S \coloneqq R \setminus \{0\}$.
|
|
%TODO
|
|
% TODO: LERNEN
|
|
|
|
|
|
% Dim k^n
|
|
$\dim(\mathfrak{k}^n)$
|
|
$ \ge n$ build chian
|
|
$\le n$ a first result in dim T ($\fp \subsetneq \fq \implies
|
|
\trdeg(\mathfrak{k}(\fq) / \mathfrak{l}) < \trdeg(\mathfrak{k}(\fp) / \mathfrak{l})$.
|
|
Thus $\codim(X,Y) \le \trdeg(\mathfrak{K}(Y) / \mathfrak{l}) -
|
|
\trdeg(\mathfrak{K}(X) / \mathfrak{l})$.
|
|
|
|
TODO
|
|
|
|
|
|
|
|
|
|
$\dim Y \ge \trdeg(\mathfrak{k}(Y) / \mathfrak{k})$: Noether normalization.
|
|
% List of proofs of HNS% Going up% TODO proof of dim Y = trdeg(K(Y) / k)
|
|
Subalgebra $\cong \mathfrak{k}[X_1,\ldots,X_d]$.
|
|
Lift chain of prime ideals using going up.
|
|
|
|
% TODO prime avoidance
|
|
|
|
|
|
Action of $\Aut(L/K)$ on prime ideals of a normal ring extension.
|
|
$A$ normal domain, $L / Q(A)$ normal field extension, $B$ int closure of $A$ in
|
|
$L$, $\fp \in \Spec A$.
|
|
Then $\Aut(L / K)$ transitively acts on $\{\fq \in \Spec B | \fq
|
|
\cap A =
|
|
\fp\}$ :
|
|
|
|
\begin{itemize}
|
|
\item
|
|
$\fq, \fr \in \Spec B$ lying over $\fp$.
|
|
\item
|
|
only need to show $\fq \subseteq \sigma(\fr)$ for some $\sigma \in G$ (Krull
|
|
going-up, no inclusions)
|
|
\item
|
|
Suppose not.
|
|
Then $x \in \fq \setminus \bigcup_{\sigma \in G} \sigma(\fr)$
|
|
(prime
|
|
aviodance)
|
|
\item
|
|
$y = \prod_{\sigma \in G} \sigma(x) \in \fq \setminus \fr$
|
|
($\fr$ prime ideal)
|
|
\item
|
|
$\exists k \in \N$ s.t.
|
|
$y^k \in K$ ($y \in L^G$)
|
|
\item
|
|
$y^k \in K \cap B = A $ ($A$ normal).
|
|
Thus $y^k \in (A \cap \fq) \setminus (A \cap \fr) = \fp \setminus \fp$.
|
|
\item
|
|
$L / K$ infinite: Apply Zorn to pairs $(M, \sigma)$ where $K \subseteq M
|
|
\subseteq L$ and $\sigma \in \Aut(M /K)$ s.t. $\sigma(\fr \cap M) = \fq \cap
|
|
M$.
|
|
\end{itemize}
|
|
|
|
|
|
Going down Krull
|
|
|
|
The ht p and trdeg ==================
|
|
|
|
Zariski-Topology, Spec, $\mathfrak{k}^n$ Residue field
|
|
$\mathfrak{k}(\fp)
|
|
\coloneqq Q(A / \fp), \mathfrak{K}(V(\fp)) \coloneqq \mathfrak{k}(\fp)$.
|
|
%TODO% TODO % TODO % TODO %% Definitions
|
|
TODO?
|
|
% Counterexamples
|
|
no going-up
|
|
% list of definitions of codim, dim, trdeg, ht
|
|
Original (Noether normalization) Artin-Tate Uncountable fields
|
|
\begin{landscape}
|
|
\section{Übersicht} {\rowcolors{2}{gray!
|
|
10}{white}
|
|
\begin{longtable}{lll}
|
|
\end{longtable}
|
|
}
|
|
|
|
\end{landscape}
|
|
|